Multiple imputation for competing risks regression with interval-censored data

ABSTRACT

We present here an extension of Pan's multiple imputation approach to Cox regression in the setting of interval-censored competing risks data. The idea is to convert interval-censored data into multiple sets of complete or right-censored data and to use partial likelihood methods to analyse them. The process is iterated, and at each step, the coefficient of interest, its variance–covariance matrix, and the baseline cumulative incidence function are updated from multiple posterior estimates derived from the Fine and Gray sub-distribution hazards regression given augmented data. Through simulation of patients at risks of failure from two causes, and following a prescheduled programme allowing for informative interval-censoring mechanisms, we show that the proposed method results in more accurate coefficient estimates as compared to the simple imputation approach. We have implemented the method in the MIICD R package, available on the CRAN website.     

The random partition masking model for interval-censored and masked competing risks data

We study the nonparametric maximum likelihood estimate (NPMLE) of the cdf or sub-distribution functions of the failure time for the failure causes in a series system. The study is motivated by a cancer research data (from the Memorial Sloan-Kettering Cancer Center) with interval-censored time and masked failure cause. The NPMLE based on this data set suggests that the existing masking models are not appropriate. We propose a new model called the random partition masking model, which does not rely on the commonly used symmetry assumption (namely, given the failure cause, the probability of observing the masked failure causes is independent of the failure time; see Flehinger et al. Inference about defects in the presence of masking, Technometrics 38 (1996), pp. 247–255). The RPM model is easier to implement in simulation studies than the existing models. We discuss the algorithms for computing the NPMLE and study its asymptotic properties. Our simulation and data analysis indicate that the NPMLE is feasible for a moderate sample size.     

Proportional hazards model for competing risks data with missing cause of failure

We consider the semiparametric proportional hazards model for the cause-specific hazard function in analysis of competing risks data with missing cause of failure. The inverse probability weighted equation and augmented inverse probability weighted equation are proposed for estimating the regression parameters in the model, and their theoretical properties are established for inference. Simulation studies demonstrate that the augmented inverse probability weighted estimator is doubly robust and the proposed method is appropriate for practical use. The simulations also compare the proposed estimators with the multiple imputation estimator of Lu and Tsiatis (2001). The application of the proposed method is illustrated using data from a bone marrow transplant study.     

Analysis of hybrid censored competing risks data

In this paper, we consider the analysis of hybrid censored competing risks data, based on Cox's latent failure time model assumptions. It is assumed that lifetime distributions of latent causes of failure follow Weibull distribution with the same shape parameter, but different scale parameters. Maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving a one-dimensional optimization problem, and we propose a fixed-point type algorithm to solve this optimization problem. Approximate MLEs have been proposed based on Taylor series expansion, and they have explicit expressions. Bayesian inference of the unknown parameters are obtained based on the assumption that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameters have Beta–Gamma priors. We propose to use Markov Chain Monte Carlo samples to compute Bayes estimates and also to construct highest posterior density credible intervals. Monte Carlo simulations are performed to investigate the performances of the different estimators, and two data sets have been analysed for illustrative purposes.     

Analysis of interval censored competing risk data with missing causes of failure using pseudo values approach

Competing risks often occur when subjects may fail from one of several mutually exclusive causes. For example, when a patient suffering a cancer may die from other cause, we are interested in the effect of a certain covariate on the probability of dying of cancer at a certain time. Several approaches have been suggested to analyse competing risk data in the presence of complete information of failure cause. In this paper, our interest is to consider the occurrence of missing causes as well as interval censored failure time. There exist no method to discuss this problem. We applied a Klein–Andersen's pseudo-value approach [Klein, JP Andersen PK. Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function. Biometrics. 2005;61:223–229] based on the estimated cumulative incidence function and a regression coefficient is estimated through a multiple imputation. We evaluate the suggested method by comparing with a complete case analysis in several simulation settings.     

Analysis of Type-II hybrid censored competing risks data

Kundu and Gupta [Analysis of hybrid life-tests in presence of competing risks. Metrica. 2007;65:159–170] provided the analysis of Type-I hybrid censored competing risks data, when the lifetime distributions of the competing cause of failures follows exponential distribution. In this paper, we consider the analysis of Type-II hybrid censored competing risks data. It is assumed that latent lifetime distributions of the competing causes of failures follow independent exponential distributions with different scale parameters. It is observed that the maximum likelihood estimators of the unknown parameters do not always exist. We propose the modified estimators of the scale parameters, which coincide with the corresponding maximum likelihood estimators when they exist, and asymptotically they are equivalent. We obtain the exact distribution of the proposed estimators. Using the exact distributions of the proposed estimators, associated confidence intervals are obtained. The asymptotic and bootstrap confidence intervals of the unknown parameters are also provided. Further, Bayesian inference of some unknown parametric functions under a very flexible Beta-Gamma prior is considered. Bayes estimators and associated credible intervals of the unknown parameters are obtained using the Monte Carlo method. Extensive Monte Carlo simulations are performed to see the effectiveness of the proposed estimators and one real data set has been analysed for the illustrative purposes. It is observed that the proposed model and the method work quite well for this data set.     

Optimal cutoffs for continuous biomarkers for survival data under competing risks

Identifying an optimal cutoff value for a continuous biomarker is often useful for medical applications. For binary outcome, commonly used cutoff finding criteria include Youden's index, classification accuracy, and the Euclidean distance to the upper left corner on the ROC curve. We extend these three criteria to accommodate censored survival time that subjected to competing risks. We provide various definitions of time-dependent true positive rate and false positive rate and estimate those quantities using nonparametric methods. In simulation studies, the Euclidean distance to the upper left corner on the ROC curve shows the best overall performance.     

Estimation of the additive hazards model with interval-censored data and missing covariates

The additive hazards model is one of the most commonly used regression models in the analysis of failure time data and many methods have been developed for its inference in various situations. However, no established estimation procedure exists when there are covariates with missing values and the observed responses are interval-censored; both types of complications arise in various settings including demographic, epidemiological, financial, medical and sociological studies. To address this deficiency, we propose several inverse probability weight-based and reweighting-based estimation procedures for the situation where covariate values are missing at random. The resulting estimators of regression model parameters are shown to be consistent and asymptotically normal. The numerical results that we report from a simulation study suggest that the proposed methods work well in practical situations. An application to a childhood cancer survival study is provided. The Canadian Journal of Statistics 48: 499–517; 2020 © 2020 Statistical Society of Canada     

Analysing competing risks data with transformation models

We present a flexible class of marginal models for the cumulative incidence function. The semiparametric transformation model is utilized in a decomposition for the marginal failure probabilities which extends previous work on Farewell's cure model. Novel estimation, inference and prediction procedures are developed, with large sample properties derived from the theory of martingales and U-statistics. A small simulation study demonstrates that the methods are appropriate for practical use. The methods are illustrated with a thorough analysis of a prostate cancer clinical trial. Simple graphical displays are used to check for the goodness of fit.     

Nonparametric estimation with recurrent competing risks data

Nonparametric estimators of component and system life distributions are developed and presented for situations where recurrent competing risks data from series systems are available. The use of recurrences of components’ failures leads to improved efficiencies in statistical inference, thereby leading to resource-efficient experimental or study designs or improved inferences about the distributions governing the event times. Finite and asymptotic properties of the estimators are obtained through simulation studies and analytically. The detrimental impact of parametric model misspecification is also vividly demonstrated, lending credence to the virtue of adopting nonparametric or semiparametric models, especially in biomedical settings. The estimators are illustrated by applying them to a data set pertaining to car repairs for vehicles that were under warranty.     

Identifiability of distributions under competing risks and complementary risks model

Let X₁,X₂,…,X_p be p random variables with cdf's F₁(x),F₂(x),…,F_p(x)respectively. Let U = min(X₁,X₂,…,X_p) and V = max(X₁,X₂,…,X_p).In this paper we study the problem of uniquely determining and estimating the marginal distributions F₁,F₂,…,F_p given the distribution of U or of V.

First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X₁'s are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X₁'s are dependent is considered next. Finally the problem of estimation is considered.     

Analysis of a design under the competing effects model with missing values

The analysis of doubly-balanced incomplete block designs (also known as 3-designs) with missing values using a competing effects model is discussed and illustrated with a numerical example.     

Modelling and analysis of recall-based competing risks data

In this paper we consider the analysis of recall-based competing risks data. The chance of an individual recalling the exact time to event depends on the time of occurrence of the event and time of observation of the individual. In particular, it is assumed that the probability of recall depends on the time elapsed since the occurrence of an event. In this study we consider the likelihood-based inference for the analysis of recall-based competing risks data. The likelihood function is constructed by incorporating the information about the probability of recall. We consider the maximum likelihood estimation of parameters. Simulation studies are carried out to examine the performance of the estimators. The proposed estimation procedure is applied to a real life data set.     

Inference for Weibull distribution under adaptive Type-I progressive hybrid censored competing risks data

In this paper, a competing risks model is considered under adaptive type-I progressive hybrid censoring scheme (AT-I PHCS). The lifetimes of the latent failure times have Weibull distributions with the same shape parameter. We investigate the maximum likelihood estimation of the parameters. Bayes estimates of the parameters are obtained based on squared error and LINEX loss functions under the assumption of independent gamma priors. We propose to apply Markov Chain Monte Carlo (MCMC) techniques to carry out a Bayesian estimation procedure and in turn calculate the credible intervals. To evaluate the performance of the estimators, a simulation study is carried out.     

On a deconvolution problem under competing risks

Bagai and Prakasa Rao [Analysis of survival data with two dependent competing risks. Biometr J. 1992;7:801–814] considered a competing risks model with two dependent risks. The two risks are initially independent but dependence arises because of the additive effect of an independent risk on the two initially independent risks. They showed that the ratio of failure rates are identifiable in the nonparametric set-up. In this paper, we consider it as a measurement error/deconvolution problem and suggest a nonparametric kernel-type estimator for the ratio of two failure rates. The local error properties of the proposed estimator are studied. Simulation studies show the efficacy of the proposed estimator.     

Fitting competing risks data to bivariate Pareto models

This paper revisits two bivariate Pareto models for fitting competing risks data. The first model is the Frank copula model, and the second one is a bivariate Pareto model introduced by Sankaran and Nair (1993 Sankaran, P. G., and N. U. Nair. 1993. A bivariate Pareto model and its applications to reliability. Naval Research Logistics 40 (7):1013–1020. doi:10.1002/1520-6750(199312)40:7%3c1013::AID-NAV3220400711%3e3.0.CO;2-7.[Crossref], [Web of Science ®] , [Google Scholar]). We discuss the identifiability issues of these models and develop the maximum likelihood estimation procedures including their computational algorithms and model-diagnostic procedures. Simulations are conducted to examine the performance of the maximum likelihood estimation. Real data are analyzed for illustration.     

On adaptive progressively Type-II censored competing risks data

In this article, a competing risks model based on exponential distributions is considered under the adaptive Type-II progressively censoring scheme introduced by Ng et al. [2009, Naval Research Logistics 56:687-698], for life testing or reliability experiment. Moreover, we assumed that some causes of failures are unknown. The maximum likelihood estimators (MLEs) of unknown parameters are established. The exact conditional and the asymptotic distributions of the obtained estimators are derived to construct the confidence intervals as well as the two different bootstraps of different unknown parameters. Under suitable priors on the unknown parameters, Bayes estimates and the corresponding two sides of Bayesian probability intervals are obtained. Also, for the purpose of evaluating the average bias and mean square error of the MLEs, and comparing the confidence intervals based on all mentioned methods, a simulation study was carried out. Finally, we present one real dataset to conduct the proposed methods.     

An additive subdistribution hazard model for competing risks data

The cumulative incidence function plays an important role in assessing its treatment and covariate effects with competing risks data. In this article, we consider an additive hazard model allowing the time-varying covariate effects for the subdistribution and propose the weighted estimating equation under the covariate-dependent censoring by fitting the Cox-type hazard model for the censoring distribution. When there exists some association between the censoring time and the covariates, the proposed coefficients’ estimations are unbiased and the large-sample properties are established. The finite-sample properties of the proposed estimators are examined in the simulation study. The proposed Cox-weighted method is applied to a competing risks dataset from a Hodgkin's disease study.     

Modeling potential time to event data with competing risks

Patients receiving radical prostatectomy are at risk of metastasis or prostate cancer related death, and often need repeated clinical evaluations to determine whether additional adjuvant or salvage therapies are needed. Since the prostate cancer is a slowly progressing disease, and these additional therapies come with significant side effects, it is important for clinical decision making purposes to estimate a patient’s risk of cancer metastasis, in the presence of a competing risk by death, under the hypothetical condition that the patient does not receive any additional therapy. In observational studies, patients may receive additional therapy by choice; the time to metastasis without any therapy is often a potential outcome and not always observed. We study the competing risks model of Fine and Gray (J Am Stat Assoc, 94:496–509, 1999) with adjustment for treatment choice by inverse probability censoring weighting (IPCW). The model can be fit using standard software for partial likelihood with double IPCW weights. The proposed methodology is used in a prostate cancer study to predict the post-prostatectomy cumulative incidence probability of cancer metastasis without additional adjuvant or salvage therapies.